The now playing snippet within the CP should stop within 1hr as well. If you're seeing it remain active longer than that, it's likely that you've navigated to another page in the CP at some point in that hour, which resets the timer.
The 1hr limit is imposed as a safeguard for the protection of the stream host -- sometimes users just leave a page open and forget about it for hours/days on end, and there's no point in letting the song status reload indefinitely, using server resources when nobody's looking at it.
This limit can be changed by the stream host (in system/streaminfo.js under his web root) but not by individual streaming clients, so I'm afraid you'd need to contact your host if you wanted to have this changed.